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What Is Math?

High school student Gracie Cunningham recently went viral with a TikTok video asking about math: how does one define math, is it real, and what is it good for? While some derided her video, mathematicians admit those are profound questions that they themselves struggle with. Is math a science, a part of science, or something on a completely different plane? The comic above illustrates, but does not answer, this conundrum. There are different schools of thought about the nature of math- some consider it a natural thing that we have discovered, while others say it was invented.

Some scholars feel very strongly that mathematical truths are “out there,” waiting to be discovered—a position known as Platonism. It takes its name from the ancient Greek thinker Plato, who imagined that mathematical truths inhabit a world of their own—not a physical world, but rather a non-physical realm of unchanging perfection; a realm that exists outside of space and time. Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist. In The Emperor’s New Mind, he wrote that there appears “to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth—a truth which has a reality of its own...”

Many mathematicians seem to support this view. The things they’ve discovered over the centuries—that there is no highest prime number; that the square root of two is an irrational number; that the number pi, when expressed as a decimal, goes on forever—seem to be eternal truths, independent of the minds that found them. If we were to one day encounter intelligent aliens from another galaxy, they would not share our language or culture, but, the Platonist would argue, they might very well have made these same mathematical discoveries.

The converse view is empiricism, in which scientists deal with things they can observe. This school of thought regards the idea that "a realm that exists outside of space and time" borders on religion and has no place in science. However, they know that math is useful for scientific observations. And there is disagreement about whether our math would be understood by alien civilizations. Read about the complexity of defining math at Smithsonian.

(Image credit: Randall Munroe at xkcd)

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